---
title: "A counterexample to Griffiths’ positivity conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 050; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "lean4"
  - "formalized"
seed_rank: 1909
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 8
lineage: ai-mathematical-reasoning
cites:
  - title: "Ample rank-two bundles on the quadric surface without Griffiths-positive metrics"
    url: "https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# A counterexample to Griffiths’ positivity conjecture

## One-sentence takeaway

OpenAI's result family 050 (Algebraic and complex geometry) claims: Constructs ample rank-two bundles on $\mathbb P^1\times\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops.

## Key ideas

- *Ample rank-two bundles on the quadric surface without Griffiths-positive metrics*: We give counterexamples to Griffiths' conjecture in rank two on the quadric surface $\mathbb P^1\times\mathbb P^1$. We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with $\mathcal O(1,1)$, are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
- Lean scope (lean/docs/050.md): Griffiths' positivity conjecture predicts that every ample holomorphic vector bundle on a smooth complex projective variety admits a smooth Hermitian metric with strictly Griffiths-positive curvature. The formalized counterexample starts with a rank-two algebraic bundle $G$ on $\mathbb P^1\times\mathbb P^1$.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: formalized in part. The repo has a scope note listing 1 Comparator challenge statement(s), and that note defines which statements are covered. Do not read the Lean coverage as the full content of every manuscript in the family.
- A grep of the repo's `lean/OAI` tree found no `sorry` and no top-level `axiom` declarations. The library was not compiled for this card, and Comparator was not run.

## Links

- Manuscript: [Ample rank-two bundles on the quadric surface without Griffiths-positive metrics](https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf)
- Lean scope: https://github.com/openai/math/blob/main/lean/docs/050.md
- Comparator statement (Ample bundles without Griffiths-positive metrics): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/QuadricBundles.lean
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math